Page 63 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 63
II: Skew Lattices
following way. First, ∧ and ∨ restricted to either A or B must be x ∧ y = y and x ∨ y = x. But
given x ∈ Ai and y ∈ Bj we set
x ∧ y = y, y ∧ x = φji(x), x ∨ y = x and y ∨ x = φji−1(y) .
The primitive skew lattice thus obtained, denoted by P[Ai, φji, Bj], has D-classes A > B, with
cosets Ai and Bj, and coset bijections φji. A left primitive skew lattice Pʹ[Ai, φji, Bj] is obtained in
dual fashion: x ∧ y = φji(x), y ∧ x = y, x ∨ y = φji−1(y) and y ∨ x = x.
A partial image of a P-graph is given in the following diagram, where the top row of dots
represents an upper D-class A of order 9 and the bottom row represents a lower D-class B of
order 6. All cosets in this example have size 3, with the members of each coset linked in the
diagram. The four arrows represent four of the six coset bijections, with arrows corresponding to
ϕ20 and ϕ02 left out.
A: • --- • --- •0 – • --- • --- •1 – • --- • --- •2
ϕ00 ϕ01 ϕ11 ϕ12
B: • --- • --- •0 – • --- • --- •1
In terms of the natural partial order, ≥, the situation looks more like
•−−•−−• •−−•−−• •−−•−−•
•−−•−−• •−−•−−•
where the dotted lines indicate the partial order relationships.
Applying Kimura factorization, every primitive skew lattice is isomorphic to the fibred
product of a left primitive skew lattice and a right primitive skew lattice over their maximal
lattice image, which here is isomorphic to 2 = {1 > 0}. This factorization coupled with the ideas
above yields:
Theorem 2.4.2. Every primitive skew lattice P has a fibred product decomposition,
P ≅ P[Ai, φji, Bj] ×P/D Pʹ[Ck, ψlk, Dl],
where (Ai, φji, Bj) and (Ck, ψlk, Dl) denote P-graphs and P/D is isomorphic to the primitive lattice
1 > 0. Both P-graphs are unique to within isomorphism of P-graphs. £
The P-graph description of right [left] primitive skew lattices can be refined as follows.
A P-graph coordinate system consists of a sextuple (I, J, C, G, µ, θ) where
61
following way. First, ∧ and ∨ restricted to either A or B must be x ∧ y = y and x ∨ y = x. But
given x ∈ Ai and y ∈ Bj we set
x ∧ y = y, y ∧ x = φji(x), x ∨ y = x and y ∨ x = φji−1(y) .
The primitive skew lattice thus obtained, denoted by P[Ai, φji, Bj], has D-classes A > B, with
cosets Ai and Bj, and coset bijections φji. A left primitive skew lattice Pʹ[Ai, φji, Bj] is obtained in
dual fashion: x ∧ y = φji(x), y ∧ x = y, x ∨ y = φji−1(y) and y ∨ x = x.
A partial image of a P-graph is given in the following diagram, where the top row of dots
represents an upper D-class A of order 9 and the bottom row represents a lower D-class B of
order 6. All cosets in this example have size 3, with the members of each coset linked in the
diagram. The four arrows represent four of the six coset bijections, with arrows corresponding to
ϕ20 and ϕ02 left out.
A: • --- • --- •0 – • --- • --- •1 – • --- • --- •2
ϕ00 ϕ01 ϕ11 ϕ12
B: • --- • --- •0 – • --- • --- •1
In terms of the natural partial order, ≥, the situation looks more like
•−−•−−• •−−•−−• •−−•−−•
•−−•−−• •−−•−−•
where the dotted lines indicate the partial order relationships.
Applying Kimura factorization, every primitive skew lattice is isomorphic to the fibred
product of a left primitive skew lattice and a right primitive skew lattice over their maximal
lattice image, which here is isomorphic to 2 = {1 > 0}. This factorization coupled with the ideas
above yields:
Theorem 2.4.2. Every primitive skew lattice P has a fibred product decomposition,
P ≅ P[Ai, φji, Bj] ×P/D Pʹ[Ck, ψlk, Dl],
where (Ai, φji, Bj) and (Ck, ψlk, Dl) denote P-graphs and P/D is isomorphic to the primitive lattice
1 > 0. Both P-graphs are unique to within isomorphism of P-graphs. £
The P-graph description of right [left] primitive skew lattices can be refined as follows.
A P-graph coordinate system consists of a sextuple (I, J, C, G, µ, θ) where
61
